| L(s) = 1 | − 5·4-s − 6·5-s − 6·11-s + 15·16-s − 8·19-s + 30·20-s + 20·25-s + 22·29-s + 46·31-s + 14·41-s + 30·44-s + 5·49-s + 36·55-s + 40·59-s − 18·61-s − 35·64-s − 12·71-s + 40·76-s − 40·79-s − 90·80-s + 24·89-s + 48·95-s − 100·100-s − 28·101-s + 10·109-s − 110·116-s − 37·121-s + ⋯ |
| L(s) = 1 | − 5/2·4-s − 2.68·5-s − 1.80·11-s + 15/4·16-s − 1.83·19-s + 6.70·20-s + 4·25-s + 4.08·29-s + 8.26·31-s + 2.18·41-s + 4.52·44-s + 5/7·49-s + 4.85·55-s + 5.20·59-s − 2.30·61-s − 4.37·64-s − 1.42·71-s + 4.58·76-s − 4.50·79-s − 10.0·80-s + 2.54·89-s + 4.92·95-s − 10·100-s − 2.78·101-s + 0.957·109-s − 10.2·116-s − 3.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 5^{10} \cdot 37^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.713284325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.713284325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{5} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 6 T + 16 T^{2} + 34 T^{3} + 24 p T^{4} + 14 p^{2} T^{5} + 24 p^{2} T^{6} + 34 p^{2} T^{7} + 16 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | \( ( 1 + T^{2} )^{5} \) |
| good | 7 | \( 1 - 5 T^{2} + 157 T^{4} - 748 T^{6} + 11922 T^{8} - 50910 T^{10} + 11922 p^{2} T^{12} - 748 p^{4} T^{14} + 157 p^{6} T^{16} - 5 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( ( 1 + 3 T + 32 T^{2} + 61 T^{3} + 470 T^{4} + 617 T^{5} + 470 p T^{6} + 61 p^{2} T^{7} + 32 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( 1 - 36 T^{2} + 456 T^{4} - 1724 T^{6} - 49056 T^{8} + 1168374 T^{10} - 49056 p^{2} T^{12} - 1724 p^{4} T^{14} + 456 p^{6} T^{16} - 36 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( 1 - 81 T^{2} + 3653 T^{4} - 114652 T^{6} + 2746906 T^{8} - 52194214 T^{10} + 2746906 p^{2} T^{12} - 114652 p^{4} T^{14} + 3653 p^{6} T^{16} - 81 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( ( 1 + 4 T + 55 T^{2} + 128 T^{3} + 1410 T^{4} + 2616 T^{5} + 1410 p T^{6} + 128 p^{2} T^{7} + 55 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 23 | \( 1 - 144 T^{2} + 10280 T^{4} - 482952 T^{6} + 16597040 T^{8} - 434849678 T^{10} + 16597040 p^{2} T^{12} - 482952 p^{4} T^{14} + 10280 p^{6} T^{16} - 144 p^{8} T^{18} + p^{10} T^{20} \) |
| 29 | \( ( 1 - 11 T + 136 T^{2} - 1025 T^{3} + 7632 T^{4} - 41283 T^{5} + 7632 p T^{6} - 1025 p^{2} T^{7} + 136 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( ( 1 - 23 T + 262 T^{2} - 2197 T^{3} + 16362 T^{4} - 102519 T^{5} + 16362 p T^{6} - 2197 p^{2} T^{7} + 262 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 41 | \( ( 1 - 7 T + 86 T^{2} + 3 T^{3} + 796 T^{4} + 17959 T^{5} + 796 p T^{6} + 3 p^{2} T^{7} + 86 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( 1 - 245 T^{2} + 30581 T^{4} - 2525036 T^{6} + 154591042 T^{8} - 7429518878 T^{10} + 154591042 p^{2} T^{12} - 2525036 p^{4} T^{14} + 30581 p^{6} T^{16} - 245 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( 1 - 238 T^{2} + 30269 T^{4} - 2575368 T^{6} + 166157154 T^{8} - 8597995284 T^{10} + 166157154 p^{2} T^{12} - 2575368 p^{4} T^{14} + 30269 p^{6} T^{16} - 238 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( 1 - 153 T^{2} + 19773 T^{4} - 1677340 T^{6} + 124372154 T^{8} - 6992655990 T^{10} + 124372154 p^{2} T^{12} - 1677340 p^{4} T^{14} + 19773 p^{6} T^{16} - 153 p^{8} T^{18} + p^{10} T^{20} \) |
| 59 | \( ( 1 - 20 T + 347 T^{2} - 4048 T^{3} + 41662 T^{4} - 341560 T^{5} + 41662 p T^{6} - 4048 p^{2} T^{7} + 347 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 61 | \( ( 1 + 9 T + 224 T^{2} + 1203 T^{3} + 19448 T^{4} + 77465 T^{5} + 19448 p T^{6} + 1203 p^{2} T^{7} + 224 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 468 T^{2} + 101704 T^{4} - 13829796 T^{6} + 1345998088 T^{8} - 101216082902 T^{10} + 1345998088 p^{2} T^{12} - 13829796 p^{4} T^{14} + 101704 p^{6} T^{16} - 468 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( ( 1 + 6 T + 171 T^{2} + 792 T^{3} + 15010 T^{4} + 65764 T^{5} + 15010 p T^{6} + 792 p^{2} T^{7} + 171 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 73 | \( 1 - 300 T^{2} + 51832 T^{4} - 6366948 T^{6} + 615079552 T^{8} - 49160084606 T^{10} + 615079552 p^{2} T^{12} - 6366948 p^{4} T^{14} + 51832 p^{6} T^{16} - 300 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 + 20 T + 382 T^{2} + 4700 T^{3} + 54438 T^{4} + 501128 T^{5} + 54438 p T^{6} + 4700 p^{2} T^{7} + 382 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 - 446 T^{2} + 102229 T^{4} - 16080104 T^{6} + 1905380754 T^{8} - 177185644148 T^{10} + 1905380754 p^{2} T^{12} - 16080104 p^{4} T^{14} + 102229 p^{6} T^{16} - 446 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( ( 1 - 12 T + 301 T^{2} - 2480 T^{3} + 44794 T^{4} - 306888 T^{5} + 44794 p T^{6} - 2480 p^{2} T^{7} + 301 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 97 | \( 1 - 473 T^{2} + 109701 T^{4} - 17966844 T^{6} + 2393147002 T^{8} - 259837694166 T^{10} + 2393147002 p^{2} T^{12} - 17966844 p^{4} T^{14} + 109701 p^{6} T^{16} - 473 p^{8} T^{18} + p^{10} T^{20} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.00137480810940827372194410762, −2.91429992809403701656967970517, −2.90492108718451514288979709820, −2.78222814885923156315006550577, −2.61260410396954960651581222428, −2.59153088052073217173869008683, −2.55567826979817689293099061639, −2.47783655816558234561235745211, −2.34360815015939876323082081045, −2.34090463162509665388290559326, −1.90637882642582063835296350766, −1.84167244400011722144110523098, −1.71066802611428041004519689702, −1.62344997431544734599408318251, −1.31677006128668609160918847260, −1.30977115654791018736151112033, −1.29808331417674363094933237587, −0.899550187736010970057749960484, −0.859819334857564487901276513036, −0.852291516328369673074908161122, −0.70630505344138837723702164522, −0.53940026411688828613713336718, −0.48394688148472140484538998146, −0.38784780962236227294971341510, −0.23303016722920847646917732496,
0.23303016722920847646917732496, 0.38784780962236227294971341510, 0.48394688148472140484538998146, 0.53940026411688828613713336718, 0.70630505344138837723702164522, 0.852291516328369673074908161122, 0.859819334857564487901276513036, 0.899550187736010970057749960484, 1.29808331417674363094933237587, 1.30977115654791018736151112033, 1.31677006128668609160918847260, 1.62344997431544734599408318251, 1.71066802611428041004519689702, 1.84167244400011722144110523098, 1.90637882642582063835296350766, 2.34090463162509665388290559326, 2.34360815015939876323082081045, 2.47783655816558234561235745211, 2.55567826979817689293099061639, 2.59153088052073217173869008683, 2.61260410396954960651581222428, 2.78222814885923156315006550577, 2.90492108718451514288979709820, 2.91429992809403701656967970517, 3.00137480810940827372194410762
Plot not available for L-functions of degree greater than 10.
